Phase diagram and critical exponents of a Potts gauge glass

Abstract

The two-dimensional q-state Potts model is subjected to a Zq symmetric disorder that allows for the existence of a Nishimori line. At q = 2, this model coincides with the ±J randombond Ising model. For q > 2, apart from the usual pure and zero-temperature fixed points, the ferro/paramagnetic phase boundary is controlled by two critical fixed points: a weak disorder point, whose universality class is that of the ferromagnetic bond-disordered Potts model, and a strong disorder point which generalizes the usual Nishimori point. We numerically study the case q = 3, tracing out the phase diagram and precisely determining the critical exponents. The universality class of the Nishimori point is inconsistent with percolation on Potts clusters.During the last decade, the study of disordered systems has attracted much interest. This is true in particular in two dimensions, where the possible types of critical behavior for the corresponding pure models can be classified using conformal field theory [1]. Recently, similar classification issues for disordered models have been addressed through the study of various random matrix ensembles [2], but many fundamental questions remain open.An important category of 2D disordered systems is given by models where the disorder couples to the local energy density. Two paradigmatic members of this class are the ±J random-bond Ising model, and the q-state ferromagnetic random-bond Potts model. The model to be studied in the present Letter can be thought of as an interpolation between these two members; we shall therefore begin by recalling some of their basic properties. The random-bond Ising model (RBIM) is defined by the energy functionalwhere the sum is over the edges of the square lattice, S i = ±1 are Ising spins, and δ(., .) is the Kronecker delta function. The random bonds take the values J ij = ±1 according to the probability distributionThe salient feature of this model is that it marries disorder with frustration, leading to the possibility of spin glass order.Its phase diagram is generally believed to be as in Fig. 1.a [3]. The boundary FP between the ferromagnetic and the paramagnetic phases is controlled by three fixed points. The attractive fixed points at either end of the phase boundary are respectively the critical point of the pure Ising model and a zero-temperature fixed point. Between these two we find the multicritical point N, intersecting the so-called Nishimori line [4] e β = (1 − p)/p .On this line, the replicated version of the model possesses a local Z 2 gauge symmetry that, among other things, allows for exactly computing the internal energy and for establishing the pairwise equality of correlation functions

Duality and Multicritical Point of Two-Dimensional Spin Glasses

“…Another problem is whether or not the whole invariant subspace (6) or (11) coincides with the phase boundary. It is dangerous to accept this identification at least below the multicritical point in the phase diagram because of the AEJ Ising model with n ¼ 2 as mentioned above.…”

“…[1][2][3][4][5][6][7][8][9][10][11] In this short note we develop a duality argument to predict the location of the multicritical point and the shape of the phase boundary in models of spin glasses on the square lattice.The first system we treat is a random Z q model with gauge symmetry which includes the AEJ Ising model and the Potts gauge glass. Following the notation of Wu and Wang, 12) the partition function is…”

Properties of the multicritical point of±JIsing spin glasses on the square lattice

“…2,8,9,12,22,23 The connection between linear susceptibility and the first moment of the correlation-function distribution is given through the fluctuation-dissipation theorem, which ͑upon invoking standard scaling arguments 29 ͒ implies the scaling relation ␥ / =2− 1 . The nonlinear susceptibility ͑nl͒ , on the other hand, can be expressed in terms of four-point correlations and products of two-point ones.…”

“…͑2͒ and ͑3͒ gives the conjectured exact location of the NP, namely, p = 0.889972¯, T / J 0 = 0.956729¯, to be referred to as CNP. In previous work, 2,7,8 approximate estimates for the location of the NP were used in the calculation of the associated critical exponents, with the overall conclusion that the transition there does not belong to the universality class of random percolation. A numerical study of correlation-function statistics at the CNP ͑Ref.…”

“…2,8,9,12,22,23 In other words, the rate of decay ͑against distance R͒ of the moments of assorted orders G n ͑R͒ of the correlation-function distribution is regulated by a set of exponents ͕ n ͖, which are not connected by a single gap exponent, as is the case for pure systems where n = n 1 . The exponent estimated via Eqs.…”